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Tropical polytopes and cellular resolutions. (English) Zbl 1134.52006

Summary: Tropical polytopes are images of polytopes in an affine space over the Puiseux series field under the degree map. This viewpoint gives rise to a family of cellular resolutions of monomial ideals that generalize the hull complex of Bayer and Sturmfels, instances of which improve upon the hull resolution in the sense of being smaller. We also suggest a new definition of a face of a tropical polytope, which has nicer properties than previous definitions; we give examples and provide many conjectures and directions for further research in this area.

MSC:

52A30 Variants of convex sets (star-shaped, (\(m, n\))-convex, etc.)
13D02 Syzygies, resolutions, complexes and commutative rings